Approximation Algorithm for Vertex Cover with Multiple Covering Constraints

Eunpyeong Hong Department of Computer Science and Information Engineering, National Taiwan University, Taipei, Taiwan [email protected] Mong-Jen Kao Department of Computer Science and Information Engineering, National Chung-Cheng University, Chiayi, Taiwan [email protected]

Abstract We consider the vertex cover problem with multiple coverage constraints in hypergraphs. In this problem, we are given a hypergraph G = (V,E) with a maximum edge size f, a cost function + w : V → Z , and edge subsets P1,P2,...,Pr of E along with covering requirements k1, k2, . . . , kr for each subset. The objective is to ﬁnd a minimum cost subset S of V such that, for each edge subset Pi, at least ki edges of it are covered by S. This problem is a basic yet general form of classical vertex cover problem and a generalization of the edge-partitioned vertex cover problem considered by Bera et al.

We present a primal-dual algorithm yielding an (f · Hr + Hr)-approximation for this problem, th where Hr is the r harmonic number. This improves over the previous ratio of (3cf log r), where c is a large constant used to ensure a low failure probability for Monte-Carlo randomized algorithms. Compared to previous result, our algorithm is deterministic and pure combinatorial, meaning that no Ellipsoid solver is required for this basic problem. Our result can be seen as a novel reinterpretation of a few classical tight results using the language of LP primal-duality.

2012 ACM Subject Classiﬁcation Mathematics of computing → Approximation algorithms

Keywords and phrases Vertex cover, multiple cover constraints, Approximation algorithm

Digital Object Identiﬁer 10.4230/LIPIcs.ISAAC.2018.43

Funding This work is supported in part by Ministry of Science and Technology (MOST), Taiwan, under Grants MOST107-2218-E-194-015-MY3 and MOST106-2221-E-001-006-MY3.

1 Introduction

The vertex cover problem is one of the most well-known and fundamental problem in graph theory and approximation algorithms. Given an undirected hypergraph G = (V,E) and a cost function w : V → Z+, the objective is to ﬁnd a minimum cost subset S ⊆ V such that any edge in E is incident to some vertex in S. This problem is known to be NP-hard, and f-approximation algorithms based on simple LP rounding and LP primal-duality are known for this problem [10], where f is the maximum size of the hyperedges. Assuming the unique game conjecture, approximating this problem to a ratio better than (f − ) is NP-hard for any > 0 [8]. The partial vertex cover problem is a natural generalization of the vertex cover problem. In this problem, we are given an additional parameter k which is called the covering requirement. The objective of this problem is to ﬁnd a minimum cost subset of V which covers at least k edges in E, i.e., at least k edges of E are incident to at least one vertex in S.

© Eunpyeong Hong and Mong-Jen Kao; licensed under Creative Commons License CC-BY 29th International Symposium on Algorithms and Computation (ISAAC 2018). Editors: Wen-Lian Hsu, Der-Tsai Lee, and Chung-Shou Liao; Article No. 43; pp. 43:1–43:11 Leibniz International Proceedings in Informatics Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany 43:2 On Vertex Cover with Multiple Covering Constraints

Various methods have been developed to obtain tight approximations for this problem. Bshouty and Burroughs [3], who ﬁrst proposed this problem, provided a 2-approximation algorithm for graphs, i.e., the case for which f = 2, using LP-rounding. Their algorithm generates |V | candidate covers each of which is constructed by guessing the most expensive vertex used in the optimal solution. Gandhi et a. [6] used the same technique to develop a primal-dual method which yields an f-approximation for hypergraphs. Mestre [9] used a more clever way to guess the most expensive vertex and improved the time complexity of the above algorithms. Fujito [5] developed an f-approximation for hypergraphs, based on a primal-dual method exploiting the property of minimal solutions. Bar-Yehuda [1] used the same property to obtain the same result using a local-ratio method. Hochbaum [7] adopted Lagrangian relaxation to get a 2-approximation on graphs. Bera et al. [2] considered a generalization of the partial vertex cover problem for which they called the partition vertex cover problem. In this problem, we are given a partition E1,E2,...,Er of the edges along with covering requirements k1, k2, . . . , kr. The objective is to ﬁnd a minimum cost vertex subset that covers at least ki edges of Ei for each 1 ≤ i ≤ r. They obtained a (6c log r)-approximation for normal graphs, where c is a large constant used for Monte-Carlo randomized algorithms to ensure low error probability. They used randomized iterative rounding on a strong LP which is derived by knapsack inequalities on the natural LP. This approach generalizes to hypergraphs with an approximation guarantee of (3cf log r). They also showed that, even for normal graph for which f = 2, it is NP-hard to approximate this problem to a ratio better than Hr, which means O(f)-approximation for this problem is unlikely to exist. Wolsey [11] proposed the submodular set cover problem, which is a general formulation to the above covering problems, and presented an H(maxS∈S g({S}))-approximation, where g is the input submodular function and S is the ground set. Chuzhoy et al. [4] presented a simpler analysis to obtain a similar result. Fujito [5] presented a primal-dual algorithm for this problem which is useful for some special cases such as the partial vertex cover problem.

Our Focus and Contributions In this paper, we consider the vertex cover problem with multiple covering constraints (VC-MCC) in hypergraphs. In this problem, we are given a hypergraph G = (V,E), a cost + function w : V → Z , and a number of covering constraints (P1, k1), (P2, k2),..., (Pr, kr), + where each Pi ⊆ E is a subset of E and ki ∈ Z is the covering requirement for Pi. The objective is to ﬁnd a minimum cost subset S ⊆ V such that, for each 1 ≤ i ≤ r, at least ki edges of Pi are covered by S. This problem is a basic yet general form of classical vertex cover and a further generalization of the edge-partitioned vertex cover problem considered in [2].

In this paper, we present a primal-dual algorithm that yields an (f·Hr+Hr)-approximation for this problem, improving over the previous ratio of (cf log r) due to [2]. Our main contribution is the following theorem.

I Theorem 1. There is a deterministic (f · Hr + Hr)-approximation algorithm for VC-MCC th which runs in polynomial time, where Hr is the r harmonic number. Compared to the previous result of (cf log r), our algorithm is deterministic and pure combinatorial, which means that our algorithm does not rely on heavy Ellipsoid LP solvers for this basic problem. Considering the lower-bound of Hr on the approximation ratio due to [2] and the well-known lower-bound of f for vertex cover, our result is much closer to the tight extent possible. E. Hong and M.-J. Kao 43:3

The novelty of this work lies in the way how we handle the dual variables. In contrast to previously known primal-dual approaches for covering problems, in which the dual variables can be handled freely, our approach manages the dual solutions carefully so that the following two criteria are met.

1. During the process, the cost of any vertex to be opened in the future must only be paid by the dual values possessed by the current unfulﬁlled covering constraints. 2. The overall dual value possessed by any unfulﬁlled covering constraint remains the same all the time. This makes the approximation guarantee of log r possible. Our result can be seen as a novel combination of the classical tight approximations with guarantees f and Hr for the covering problem, using the language of LP primal-duality. Our ingredient includes the strong LP relaxation due to [2], which is derived by applying Knapsack-cover inequalities to the natural LP. We would like to remark, however, that the usage of strong LP relaxation in our result is not a necessity but rather a better and more intuitive exposition of our ideas on how the dual variables can be managed, and obtaining the same result using natural LP is possible.

Organization of this paper The rest of this paper is organized as follows. In Section 2, we deﬁne the notations we will be using throughout this paper and introduce the strong LP formulations. We present our approximation algorithm in Section 3 and conclude with future directions in Section 4.

2 Preliminary

We use G = (V,E) to denote a hypergraph G with a vertex set V and an edge set E ⊆ 2V . Note that, under this notion, any edge e ∈ E is a subset of V that consists the incident vertices of the edge e. We use fG to denote the maximum cardinality of the edges in G, i.e., fG = maxe∈E |e|. The subscript G is omitted when no ambiguity is there in the context. For any edge subset M ⊆ E and any vertex v ∈ V , we use M(v) to denote the set of edges in M that are incident to v, i.e., M(v) := {e ∈ M : v ∈ e}. For any subset A ⊆ V , we use M(A) to denote the set of edges in M that are incident to the vertices in A, i.e., S M(A) = v∈A M(v).

Vertex Cover with Multiple Covering Constraints

In this problem, we are given a hypergraph G = (V,E), a cost function w : V → Z+, and a number of covering constraints (P1, k1), (P2, k2),..., (Pr, kr), where for each 1 ≤ i ≤ r, + 1 Pi ⊆ E is a subset of E and ki ∈ Z is the covering requirement for Pi to be fulﬁlled. The objective of this problem is to ﬁnd a vertex subset S ⊆ V of minimum cost such that

Pi(S) ≥ ki for each 1 ≤ i ≤ r.

Intuitively, this problem asks for a minimum cost subset such that in each Pi, at least ki edges are covered. A natural LP relaxation for this problem is given in Figure 1. We have two sets of indicator variables in this LP formulation: For each v ∈ V , xv denotes the inclusion of v into the cover and ye for each e ∈ E indicates the coverage of e by the

1 Without loss of generality, we assume that ki ≤ |Pi| for all 1 ≤ i ≤ r.

I S A AC 2 0 1 8 43:4 On Vertex Cover with Multiple Covering Constraints

X min wvxv LP-(N) v∈V X s.t. xv ≥ ye, ∀e ∈ E v∈e X ye ≥ ki, ∀1 ≤ i ≤ r

e∈Pi

xv ≥ 0, ∀v ∈ V

1 ≥ ye ≥ 0, ∀e ∈ E.

Figure 1 A natural LP relaxation for VC-MCC.

vertices chosen in the cover. The ﬁrst inequality models the coverage of each edge e ∈ E and the second inequality models the covering requirement for each (Pi, ki), 1 ≤ i ≤ r. However, the integrality gap of the natural LP can be arbitrarily large. This is illustrated by the following simple example. Consider a star with d + 1 vertices. Suppose that the cost of every vertex is 1 and we only have one constraint consisting of all edges with covering requirement 1. The optimal integral cost for this example is 1 while its optimal fractional cost is 1/d, resulting a gap of d which can be arbitrarily large.

A Strong LP Relaxation Instead of using the natural LP relaxation, we use a strong LP relaxation due to [2], which is derived by applying Knapsack-cover inequalities to the natural LP given above. For any vertex subset A ⊆ V and any 1 ≤ i ≤ r, deﬁne

ki(A) := max ki − Pi(A) , 0 .

Intuitively, ki(A) denotes the residue covering requirement to be fulﬁlled for Pi, if the vertex set A were already chosen as part of the cover. For any vertex v ∈ V \ A, deﬁne

βi(v, A) := min Pi(v) \ Pi(A) , ki(A) .

Intuitively, βi(v, A) is the amount of covering requirement v can be fulﬁlled for Pi if A is already chosen as part of the cover. Clearly, βi(v, A) will be either ki(A) or the number of incident edges of v in Pi \ Pi(A), which is |Pi(v) \ Pi(A)|. The strong LP relaxation we consider is as follows:

X min wv · xv LP-(S) v∈V X s.t. βi(v, A) · xv ≥ ki(A), ∀1 ≤ i ≤ r, ∀A ⊆ V v∈V \A

xv ≥ 0, ∀v ∈ V.

To see that LP-(S) gives a valid relaxation for VC-MCC, consider any feasible integral solution xˆ. It suﬃces to show that xˆ is also contained in the feasible region of LP-(S). E. Hong and M.-J. Kao 43:5

Consider an arbitrary subset A of V and any constraint 1 ≤ i ≤ r. Clearly, xˆ must remain feasible even if the vertices of A were already chosen as part of the cover in advance for free. Hence, the number of edges xˆ covers for Pi is at least ki(A), and the inequality X βi(v, A) · xv ≥ ki(A) v∈V \A must hold. To see that LP-(S) is indeed a stronger relaxation than LP-(N), let us consider the simple star example and the inequality with respect to A = ∅. Clearly, β1(v, A) = 1 for all vertices P v in this star. As a result, we have a constraint v∈V xv ≥ 1, and the optimal fractional solution will also be 1.

The Dual LP for LP-(S) In this paper we will be working around the dual LP of LP-(S), which is given as follows.

X max ki(A) · zi,A LP-Dual-(S) 1≤i≤r, A⊆V X s.t. βi(v, A) · zi,A ≤ wv, ∀v ∈ V, (*) 1≤i≤r, A⊆V \{v}

zi,A ≥ 0, ∀1 ≤ i ≤ r, ∀A ⊆ V

3 Our Approximation Algorithm for VC-MCC

In this section, we present our approximation algorithm for VC-MCC. Given an instance Π = G = (V,E), w, (Pi, ki) 1≤i≤r of VC-MCC, the algorithm will compute a series of feasible LP solutions of Π to LP-Dual-(S). During this process, a feasible cover for Π will gradually be formed. The approximation guarantee is then established by comparing the cost of the cover to the values of the dual solutions the algorithm computes. In the following section we describe the algorithm in details. In §3.2 we establish the approximation guarantee.

3.1 The Algorithm The algorithm takes as input an instance Π of VC-MCC and outputs a feasible cover S for Π. Initially, S is set to be an empty set. In addition, the algorithm will maintain a set K which contains the set of covering constraints that have not been satisﬁed yet. The set K is initialized to be {1, 2, . . . , r}. In the following, we ﬁrst describe our primal-dual process. Then we describe how this primal-dual process can be transformed into a polynomial-time algorithm. Our primal-dual process, denoted PD-VC-MCC, starts with a trivial dual solution for which zi,A = 0 for all 1 ≤ i ≤ r and all A ⊆ V . In each iteration, it proceeds as follows:

1. It raises zi,S at the rate of 1/ki(S) for all i ∈ K until the Inequality (*) of some vertex in LP-Dual-(S), say, v ∈ V \ S, becomes tight. Then it adds the vertex v to the set S.

I S A AC 2 0 1 8 43:6 On Vertex Cover with Multiple Covering Constraints

Input An instance Π of VC-MCC Output A feasible cover S

1. S ← ∅, K ← {1, . . . , r} and zi,A ← 0 for all 1 ≤ i ≤ r and all A ⊆ V . 2. Repeat until K becomes an empty set. a. Raise zi,S for all i ∈ K simultaneously at the rate of 1/ki(S) until the inequality (*) in LP-Dual-(S) for some v ∈ V \ S becomes tight. b. Add the vertex v into S. c. For every i ∈ K such that ki(S) = 0, K ← K \{i} and zi,A ← 0 for all A ⊆ V . 3. Return S

Figure 2 A formal description of our primal-dual process PD-VC-MCC.

2. For each constraint i ∈ K with ki(S) becoming zero after v is added to S, our primal-dual process:

a. sets zi,A to be zero for all A ⊆ V and b. removes i from K. This process repeats until the set K becomes empty. Then S is returned as the approximate solution. A high-level pseudo-code of this primal-dual process is given in Figure 2 for further reference.

We remark that, the step 2.(a) above of resetting zi,A to zero for all A ⊆ V when i is to be removed from K is very important and is the key to obtain a guarantee of Hr. The reason is that it allows the overall contribution of the remaining covering constraints to remain balanced. Our approximation algorithm, denoted Approx-VC-MCC, mimics the operations of the above primal-dual process. Instead of maintaining the dual variables, it keeps track of the slack of the Inequality (*) in LP-Dual-(S) for each vertex, i.e., the amount before it becomes tight.

For each v ∈ V , let wˆv denote the slack of the vertex constraint v before it becomes tight. Initially, wˆv is set to be wv. We need a notion that reﬂects the raising process of the dual variables. For each v ∈ V and A ⊆ V \{v}, deﬁne

X βi(v, A) s(v, A) := . ki(A) i∈K

Intuitively, s(v, A) denotes the speed for which wˆv will decrease if we raised the dual variables zi,A at the speed of 1/ki(A) for all i ∈ K. Furthermore, in order for the update of wˆv for each v ∈ V to proceed, we use Φv,i to denote the contribution of dual variables zi,A for all possible A towards wˆv. Φv,i is initialized to be zero for all v ∈ V and 1 ≤ i ≤ r.

Now we formally describe our approximation algorithm. In each iteration, the algorithm ﬁnds the among the vertices in V \ S the one with the smallest ratio of wˆv/s(v, A). Formally speaking, it computes

wˆu wˆv v = arg min and tv = . u∈V \S s(u, A) s(v, A) E. Hong and M.-J. Kao 43:7

Intuitively, v is the ﬁrst vertex constraint to become tight in this iteration in the primal-dual process and tv is the corresponding amount of time it takes. Then the algorithm proceeds as follows:

1. For each u ∈ V \ S, the algorithm:

a. updates wû by setting wû ← wû − s(u, S) · tv. βi(u,S) b. update the contribution Φu,i for each i ∈ K by setting Φu,i ← Φu,i + · tv. ki(S) 2. Add v to the set S.

3. For each i ∈ K such that ki(S) is zero, the algorithm does the following: a. Update wû for all u ∈ V \ S by setting wû ← wû + Φu,i. b. Remove i from K.

The algorithm repeats until the set K becomes empty. Then S is returned as the approximate solution.

3.2 Analysis In this section, we provide the analysis of our approximation algorithm Approx-VC-MCC and prove Theorem 1. First we show that our algorithm always terminates and returns a feasible cover. Then we establish the approximation guarantee.

Feasibility of algorithm Approx-VC-MCC We ﬁrst establish the feasibility of our primal-dual process. Then we argue that algorithm Approx-VC-MCC does mimic the execution of this process and runs in polynomial time.

I Lemma 2. The primal-dual process PD-VC-MCC always terminates and returns a feasible cover.

Proof. Since PD-VC-MCC only terminates when the set K becomes empty and it ﬁnds a feasible cover, it suﬃces to argue that PD-VC-MCC always terminates, provided that there is a feasible cover for the input instance. Assume for contradiction that the input instance has a feasible solution but PD-VC-MCC does not terminate. Consider the set S the process currently has. The process runs eternally since no vertex v ∈ V \ S becomes tight as zi,S is constantly raising for all i ∈ K. This implies that βi(v, S) = |Pi(v) \ Pi(S)| = 0 for all v ∈ V \ S and all i ∈ K. This means that all the edges have already been covered by S, a contradiction. J

To see that algorithm Approx-VC-MCC simulates the execution of PD-VC-MCC, it suﬃces to observe that P wˆv records the slack wv − 1≤i≤r, A⊆V \{v} βi(v, A) · zi,A of the constraint (*) for all v ∈ V \ S during all iterations, P Φv,i keeps track of the value A⊆V \{v} βi(v, A) · zi,A so that it can be used to reﬂect the operation of resetting zi,A to zero for i that is about to be removed from K.

We have the following lemma.

I Lemma 3. Algorithm Approx-VC-MCC mimics the execution of primal-dual process PD-VC-MCC and runs in polynomial time.

Lemma 2 and Lemma 3 establish the feasibility of algorithm Approx-VC-MCC.

I S A AC 2 0 1 8 43:8 On Vertex Cover with Multiple Covering Constraints

Approximation Guarantee To establish the approximation guarantee, we compare the cost of the solution our algorithm returns to the values of the dual solutions our primal-dual process maintains, which will be valid lower-bounds for the cost of optimal solutions by the weak LP duality. Let S = {v1, v2, . . . , vm} denote the cover returned by the algorithm, where the indices of the vertices denote the order for which they are added to the set S. For any 0 ≤ j ≤ m, we use Aj to denote the set of the ﬁrst j vertices that are added to S, i.e., Aj := {v1, v2, . . . , vj}. Without loss of generality, we also assume that the covering constraints P1,P2,...,Pr are fulﬁlled by the algorithm in this order.

For any 1 ≤ i ≤ r, let π(i) denote the index for which the inclusion of vπ(i) into S fulﬁlls Pi. Consider the moment when Pi is just fulﬁlled, i.e., when i was removed from K by the (i) algorithm, and zi,A has not yet been reset. Let zˆ denote the dual solution the algorithm maintains at this moment, and Val zˆ(i) denote the objective value of zˆ(i). It follows that

X (i) X X (i) Val zˆ(i) := k (A) · zˆ = k (A ) · zˆ , (1) t t,A t j t,Aj 1≤t≤r, A⊆V i≤t≤r 0≤j<π(i)

(i) where the second equality holds since our algorithm resets zˆt,A to zero for all 1 ≤ t < i and all A ⊆ V . In the above equality we write Val zˆ(i) as the sum of dual values each unfulﬁlled covering requirement possesses. The following lemma says that the dual value possessed by each unfulﬁlled constraint is the same.

I Lemma 4. For any 1 ≤ i ≤ r and any t1, t2 with i ≤ t1 6= t2 ≤ r, we have X (i) X (i) k (A ) · zˆ = k (A ) · zˆ . t1 j t1,Aj t2 j t2,Aj 0≤j<π(i) 0≤j<π(i) Proof. This lemma follows directly from the way our primal-dual approach handles the dual variables. Since ki(S) only changes when a new vertex becomes tight and since we always raise zi,S for each i ∈ K at the rate of 1/ki(S), the total dual value possessed by any Pi with i ∈ K will be the same. J

In the following we analyze the cost of S and relate it to the dual values of zˆ(i) the algorithm maintains for all 1 ≤ i ≤ r. Consider a vertex v ∈ S and the moment when v just becomes tight. Suppose that at that time, the algorithm has already fulﬁlled t covering constraints. Then, from the Inequality (*) of LP-Dual-(S), it follows that X X wv = βi(v, A) · zi,A = βi(v, A) · zi,A, 1≤i≤r, A⊆V \{v} t

where the second equality holds since, by design, our algorithm has already reset zi,A to zero for all 1 ≤ i ≤ t and all A ⊆ V before v becomes tight. P For each t < i ≤ r, deﬁne Φv,i := A⊆V \{v} βi(v, A) · zi,A. Then we have X wv = Φv,i. t

Intuitively, Φv,i is the share for which the covering constraint i contributes towards the cost of vertex v. We will charge the cost of v to the covering constraints Pi for all t < i ≤ r, each of which gets a charge of Φv,i. E. Hong and M.-J. Kao 43:9

For each 1 ≤ i ≤ r, let cost(Pi) denote the total charge Pi receives from the vertices in S. (i) We have the following lemma, which bounds cost(Pi) using the dual value it possesses in zˆ .

I Lemma 5. For any 1 ≤ i ≤ r, we have

X (i) cost(P ) ≤ (f + 1) · k (A ) · zˆ . i i j i,Aj 0≤j<π(i)

Proof. From the deﬁnition of cost(Pi), only v1, v2, . . . , vπ(i) will charge Pi, and it follows that

X X X (i) cost(Pi) = Φvt,i = βi(vt,A) · zˆi,A

1≤t≤π(i) 1≤t≤π(i) A⊆V \{vt}

X X (i) = β (v ,A ) · zˆ i t j i,Aj 1≤t≤π(i) 0≤j

X X (i) = β (v ,A ) · zˆ . i t j i,Aj 0≤j<π(i) j

Hence, to prove this lemma, it suﬃces to show that

X X (i) X (i) β (v ,A ) · zˆ ≤ (f + 1) · k (A ) · zˆ . (2) i t j i,Aj i j i,Aj 0≤j<π(i) j

To prove Ineq. (2), we ﬁrst prove the following inequality:

X X (i) X (i) β (v ,A ) · zˆ ≤ f · k (A ) · zˆ (3) i t j i,Aj i j i,Aj 0≤j<π(i) j

Compare the l.h.s. and the r.h.s. of (3), it suﬃces to argue that X βi(vt,Aj) ≤ f · ki(Aj) for all 0 ≤ j < π(i). j

Consider any ﬁxed j with 0 ≤ j < π(i) and any t with j < t < π(i). Since vt is not the

vertex whose inclusion into S fulﬁlls Pi, it follows that Pi(vt) \ Pi(Aj) < ki(Aj), and

hence βi(vt,Aj) = Pi(vt) \ Pi(Aj) . Furthermore, under the condition that Aj has already been chosen, the inclusion of

{vj+1, vj+2, . . . , vπ(i)−1} into S does not fulﬁll Pi.

S This implies that j

X X βi(vt,Aj) = Pi(vt) \ Pi(Aj) j

[ ≤ f · (Pi(vt) \ Pi(Aj)) < f · ki(Aj),

where the second last inequality holds since the size of each hyperedge is at most f. This proves Ineq. (3).

I S A AC 2 0 1 8 43:10 On Vertex Cover with Multiple Covering Constraints

Second, observe that the following inequality holds

X (i) X (i) β (v ,A ) · zˆ ≤ k (A ) · zˆ , (4) i π(i) j i,Aj i j i,Aj 0≤j<π(i) 0≤j<π(i)

since βi(vπ(i),Aj) ≤ ki(Aj) by the deﬁnition of βi(vπ(i),Aj). From Ineq. (3) and Ineq. (4), the Inequality (2) is proved and this lemma holds. J In the following we establish the approximation guarantee of our algorithm.

I Lemma 6.

cost(S) ≤ (f + 1) · Hr · OPT,

th where Hr is the r harmonic number and OPT is the cost of any optimal solution.

Proof. By Lemma 5 and the deﬁnition of cost(Pi), we have

X X X (i) cost(S) = cost(P ) ≤ (f + 1) · k (A ) · zˆ . i i j i,Aj 1≤i≤r 1≤i≤r 0≤j<π(i)

By Lemma 4, for each 1 ≤ i ≤ r, we have

X (i) 1 (i) ki(Aj) · zˆ = · Val zˆ . i,Aj r − i + 1 0≤j<π(i)

Since each zˆ(i) is a feasible dual solution for LP-Dual-(S), we have Val zˆ(i) ≤ OPT for all 1 ≤ i ≤ r, and X 1 cost(S) ≤ (f + 1) · · OPT ≤ (f + 1) · H · OPT r − i + 1 r 1≤i≤r

as claimed. J

4 Conclusion

We conclude with future directions and open problems. First, considering the lower-bounds of Hr and f for this problem, our (f · Hr + Hr)-approximation ratio has an extra Hr factor in it. However, it seems unclear how this excess Hr factor can be dropped. Although the approaches of [5, 6, 9] can be used to obtain tight f-approximation for the partial vertex cover problem, it seems diﬃcult to adopt their techniques to our problem. The reason is that, when multiple covering constraints exist, it seems intricate how the key properties of their approaches can be ensured simultaneously for each covering constraint. We believe that this would be an interesting direction to explore. Second, clarifying the exact lower bound of approximation ratio for this problem is also interesting. For now, max(Hr, f) is what we only know.

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